S is the surface z = xy in Euclidean 3-space.

Find all straight lines lying in S.

The parametric representation of a line in 3-space is x=x0+a*t, y=y0+b*t, z=z0+c*t, where (x0,y0,z0) is a point on the line and <a,b,c> a vector parallel to the line. Since (x0,y0,z0) is on the line, it is on the surface and z0=x0*y0.

Substituting the parametric equations into the equation for the surface gives

z0+c*t = (x0+a*t)*(y0+b*t)
x0*y0+c*t = x0*y0 + (a*y0+b*x0)*t + a*b*t^2
c*t = (a*y0+b*x0)*t + a*b*t^2
c = a*y0+b*x0 + a*b*t

In order for c to be constant, a or b has to be equal to 0. Both cannot be equal to 0 or else it would only be one point (x0,y0,z0). Hence the lines are:

x=x0, y=y0+b*t, z=z0+c*t; where c=b*x0
x=x0+a*t, y=y0, z=z0+c*t; where c=a*y0

Charlie's solution only includes the lines that pass through the origin. This general solution includes lines that pass through all points of S.

Posted by np_rt on 2004-02-24 04:59:38